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76-519: Technically, the ferry is a reaction ferry , which is propelled by the current of the water. An overhead cable is suspended from towers anchored on either bank of the river. The ferry is attached to this overhead cable by bridle cables and pulleys. To operate the ferry, it is angled into the current, causing the force of the current to move the ferry across the river. 51°29′53″N 9°36′19″E  /  51.498184°N 9.605291°E  / 51.498184; 9.605291 This ferry article

152-426: A {\displaystyle {\mathfrak {a}}} . Vectors are usually shown in graphs or other diagrams as arrows (directed line segments ), as illustrated in the figure. Here, the point A is called the origin , tail , base , or initial point , and the point B is called the head , tip , endpoint , terminal point or final point . The length of the arrow is proportional to the vector's magnitude , while

228-473: A z = a x i + a y j + a z k . {\displaystyle \mathbf {a} =\mathbf {a} _{x}+\mathbf {a} _{y}+\mathbf {a} _{z}=a_{x}{\mathbf {i} }+a_{y}{\mathbf {j} }+a_{z}{\mathbf {k} }.} The notation e i is compatible with the index notation and the summation convention commonly used in higher level mathematics, physics, and engineering. As explained above ,

304-854: A 1 a 2 a 3 ] = [ a 1   a 2   a 3 ] T . {\displaystyle \mathbf {a} ={\begin{bmatrix}a_{1}\\a_{2}\\a_{3}\\\end{bmatrix}}=[a_{1}\ a_{2}\ a_{3}]^{\operatorname {T} }.} Another way to represent a vector in n -dimensions is to introduce the standard basis vectors. For instance, in three dimensions, there are three of them: e 1 = ( 1 , 0 , 0 ) ,   e 2 = ( 0 , 1 , 0 ) ,   e 3 = ( 0 , 0 , 1 ) . {\displaystyle {\mathbf {e} }_{1}=(1,0,0),\ {\mathbf {e} }_{2}=(0,1,0),\ {\mathbf {e} }_{3}=(0,0,1).} These have

380-412: A 1 , a 2 , a 3 ) . {\displaystyle \mathbf {a} =(a_{1},a_{2},a_{3}).} also written, a = ( a x , a y , a z ) . {\displaystyle \mathbf {a} =(a_{x},a_{y},a_{z}).} This can be generalised to n-dimensional Euclidean space (or R ). a = (

456-397: A 1 , a 2 , a 3 , ⋯ , a n − 1 , a n ) . {\displaystyle \mathbf {a} =(a_{1},a_{2},a_{3},\cdots ,a_{n-1},a_{n}).} These numbers are often arranged into a column vector or row vector , particularly when dealing with matrices , as follows: a = [

532-598: A 3 ( 0 , 0 , 1 ) ,   {\displaystyle \mathbf {a} =(a_{1},a_{2},a_{3})=a_{1}(1,0,0)+a_{2}(0,1,0)+a_{3}(0,0,1),\ } or a = a 1 + a 2 + a 3 = a 1 e 1 + a 2 e 2 + a 3 e 3 , {\displaystyle \mathbf {a} =\mathbf {a} _{1}+\mathbf {a} _{2}+\mathbf {a} _{3}=a_{1}{\mathbf {e} }_{1}+a_{2}{\mathbf {e} }_{2}+a_{3}{\mathbf {e} }_{3},} where

608-721: A 1 , a 2 , a 3 are called the vector components (or vector projections ) of a on the basis vectors or, equivalently, on the corresponding Cartesian axes x , y , and z (see figure), while a 1 , a 2 , a 3 are the respective scalar components (or scalar projections). In introductory physics textbooks, the standard basis vectors are often denoted i , j , k {\displaystyle \mathbf {i} ,\mathbf {j} ,\mathbf {k} } instead (or x ^ , y ^ , z ^ {\displaystyle \mathbf {\hat {x}} ,\mathbf {\hat {y}} ,\mathbf {\hat {z}} } , in which

684-412: A . ( Uppercase letters are typically used to represent matrices .) Other conventions include a → {\displaystyle {\vec {a}}} or a , especially in handwriting. Alternatively, some use a tilde (~) or a wavy underline drawn beneath the symbol, e.g. a ∼ {\displaystyle {\underset {^{\sim }}{a}}} , which

760-418: A directed line segment , or arrow, in a Euclidean space . In pure mathematics , a vector is defined more generally as any element of a vector space . In this context, vectors are abstract entities which may or may not be characterized by a magnitude and a direction. This generalized definition implies that the above-mentioned geometric entities are a special kind of abstract vectors, as they are elements of

836-600: A Euclidean space E is defined as a set to which is associated an inner product space of finite dimension over the reals E → , {\displaystyle {\overrightarrow {E}},} and a group action of the additive group of E → , {\displaystyle {\overrightarrow {E}},} which is free and transitive (See Affine space for details of this construction). The elements of E → {\displaystyle {\overrightarrow {E}}} are called translations . It has been proven that

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912-442: A basis does not affect the properties of a vector or its behaviour under transformations. A vector can also be broken up with respect to "non-fixed" basis vectors that change their orientation as a function of time or space. For example, a vector in three-dimensional space can be decomposed with respect to two axes, respectively normal , and tangent to a surface (see figure). Moreover, the radial and tangential components of

988-728: A condition may be emphasized calling the result a bound vector . When only the magnitude and direction of the vector matter, and the particular initial or terminal points are of no importance, the vector is called a free vector . The distinction between bound and free vectors is especially relevant in mechanics, where a force applied to a body has a point of contact (see resultant force and couple ). Two arrows A B ⟶ {\displaystyle {\stackrel {\,\longrightarrow }{AB}}} and A ′ B ′ ⟶ {\displaystyle {\stackrel {\,\longrightarrow }{A'B'}}} in space represent

1064-414: A convenient algebraic characterization of both angle (a function of the dot product between any two non-zero vectors) and length (the square root of the dot product of a vector by itself). In three dimensions, it is further possible to define the cross product , which supplies an algebraic characterization of the area and orientation in space of the parallelogram defined by two vectors (used as sides of

1140-436: A convenient numerical fashion. For example, the sum of the two (free) vectors (1, 2, 3) and (−2, 0, 4) is the (free) vector ( 1 , 2 , 3 ) + ( − 2 , 0 , 4 ) = ( 1 − 2 , 2 + 0 , 3 + 4 ) = ( − 1 , 2 , 7 ) . {\displaystyle (1,2,3)+(-2,0,4)=(1-2,2+0,3+4)=(-1,2,7)\,.} In

1216-425: A corresponding bound vector, in this sense, whose initial point has the coordinates of the origin O = (0, 0, 0) . It is then determined by the coordinates of that bound vector's terminal point. Thus the free vector represented by (1, 0, 0) is a vector of unit length—pointing along the direction of the positive x -axis. This coordinate representation of free vectors allows their algebraic features to be expressed in

1292-404: A land yacht and the moving fluid is the water current rather than the wind. In the case of a reaction ferry with an anchored tether, the analogy can also be to a kite . In both cases the ferry's hull itself represents a sail and is angled to the apparent water current in order to generate lift in the same way a sail is set at an angle to the apparent wind . With an overhead cable stretched across

1368-417: A period of more than 200 years. About a dozen people contributed significantly to its development. In 1835, Giusto Bellavitis abstracted the basic idea when he established the concept of equipollence . Working in a Euclidean plane, he made equipollent any pair of parallel line segments of the same length and orientation. Essentially, he realized an equivalence relation on the pairs of points (bipoints) in

1444-495: A river at right angles to the current, the ferry is, in sailing terminology, sailing on a reach with the true current exactly at right angles to the direction of crossing. For the anchored-tether type ferry this is valid when the tether is parallel to the current, near the middle of crossing. In sailing, the speed is governed by the lift-to-drag ratios (L/D) of the sail and the hull including centerboard or keel and rudder. For reaction ferries, L/D ratios also apply except that one

1520-425: A space with no notion of length or angle. In physics, as well as mathematics, a vector is often identified with a tuple of components, or list of numbers, that act as scalar coefficients for a set of basis vectors . When the basis is transformed, for example by rotation or stretching, then the components of any vector in terms of that basis also transform in an opposite sense. The vector itself has not changed, but

1596-484: A special kind of vector space called Euclidean space . This particular article is about vectors strictly defined as arrows in Euclidean space. When it becomes necessary to distinguish these special vectors from vectors as defined in pure mathematics, they are sometimes referred to as geometric , spatial , or Euclidean vectors. A Euclidean vector may possess a definite initial point and terminal point ; such

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1672-406: A vector is often described by a set of vector components that add up to form the given vector. Typically, these components are the projections of the vector on a set of mutually perpendicular reference axes (basis vectors). The vector is said to be decomposed or resolved with respect to that set. The decomposition or resolution of a vector into components is not unique, because it depends on

1748-726: A vector is one type of tensor . In pure mathematics , a vector is any element of a vector space over some field and is often represented as a coordinate vector . The vectors described in this article are a very special case of this general definition, because they are contravariant with respect to the ambient space. Contravariance captures the physical intuition behind the idea that a vector has "magnitude and direction". Vectors are usually denoted in lowercase boldface, as in u {\displaystyle \mathbf {u} } , v {\displaystyle \mathbf {v} } and w {\displaystyle \mathbf {w} } , or in lowercase italic boldface, as in

1824-494: A vector pointing into and behind the diagram. These can be thought of as viewing the tip of an arrow head on and viewing the flights of an arrow from the back. In order to calculate with vectors, the graphical representation may be too cumbersome. Vectors in an n -dimensional Euclidean space can be represented as coordinate vectors in a Cartesian coordinate system . The endpoint of a vector can be identified with an ordered list of n real numbers ( n - tuple ). These numbers are

1900-853: A vector relate to the radius of rotation of an object. The former is parallel to the radius and the latter is orthogonal to it. In these cases, each of the components may be in turn decomposed with respect to a fixed coordinate system or basis set (e.g., a global coordinate system, or inertial reference frame ). The following section uses the Cartesian coordinate system with basis vectors e 1 = ( 1 , 0 , 0 ) ,   e 2 = ( 0 , 1 , 0 ) ,   e 3 = ( 0 , 0 , 1 ) {\displaystyle {\mathbf {e} }_{1}=(1,0,0),\ {\mathbf {e} }_{2}=(0,1,0),\ {\mathbf {e} }_{3}=(0,0,1)} and assumes that all vectors have

1976-471: Is Minkowski space (which is important to our understanding of special relativity ). However, it is not always possible or desirable to define the length of a vector. This more general type of spatial vector is the subject of vector spaces (for free vectors) and affine spaces (for bound vectors, as each represented by an ordered pair of "points"). One physical example comes from thermodynamics , where many quantities of interest can be considered vectors in

2052-416: Is −15 N. In either case, the magnitude of the vector is 15 N. Likewise, the vector representation of a displacement Δ s of 4 meters would be 4 m or −4 m, depending on its direction, and its magnitude would be 4 m regardless. Vectors are fundamental in the physical sciences. They can be used to represent any quantity that has magnitude, has direction, and which adheres to

2128-405: Is a stub . You can help Misplaced Pages by expanding it . This Hesse location article is a stub . You can help Misplaced Pages by expanding it . This Lower Saxony location article is a stub . You can help Misplaced Pages by expanding it . Reaction ferry A reaction ferry is a cable ferry that uses the reaction of the current of a river against a fixed tether to propel the vessel across

2204-417: Is a convention for indicating boldface type. If the vector represents a directed distance or displacement from a point A to a point B (see figure), it can also be denoted as A B ⟶ {\displaystyle {\stackrel {\longrightarrow }{AB}}} or AB . In German literature, it was especially common to represent vectors with small fraktur letters such as

2280-407: Is a vector-valued physical quantity , including units of measurement and possibly a support , formulated as a directed line segment . A vector is frequently depicted graphically as an arrow connecting an initial point A with a terminal point B , and denoted by A B ⟶ . {\textstyle {\stackrel {\longrightarrow }{AB}}.} A vector

2356-465: Is essentially the modern system of vector analysis. In 1901, Edwin Bidwell Wilson published Vector Analysis , adapted from Gibbs's lectures, which banished any mention of quaternions in the development of vector calculus. In physics and engineering , a vector is typically regarded as a geometric entity characterized by a magnitude and a relative direction . It is formally defined as

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2432-431: Is often presented as the standard Euclidean space of dimension n . This is motivated by the fact that every Euclidean space of dimension n is isomorphic to the Euclidean space R n . {\displaystyle \mathbb {R} ^{n}.} More precisely, given such a Euclidean space, one may choose any point O as an origin . By Gram–Schmidt process , one may also find an orthonormal basis of

2508-409: Is really a three-dimensional situation. R can be resolved in a drag component directly downstream and a component in the direction of crossing, the thrust T which drives the ferry. This is balanced by the opposing drag of the traveller pulleys. The amount of lift required is set by the angle of incidence of the ferry to the apparent current (here 10°), often done with a rudder (not shown). In the figure

2584-786: Is reputed to have been designed by Leonardo da Vinci . A number of reaction ferries operate: Four passenger ferries cross the Rhine in Basel . Three such ferries cross the Aare in Bern . A small traditional ferry, the last on this river, crosses the Doubs . The Hampton Loade Ferry , which carried passengers only, crossed the River Severn at Hampton Loade in Shropshire until 2017. It

2660-408: Is required to initially pull the cable and also to manoeuvre, especially during the turning of the tide. The ferry may consist of a single hull, or two pontoons with a deck bridging them. Some ferries carry only passengers, whilst others carry road vehicles, with some examples carrying up to 12 cars. A reaction ferry operates as a sailing craft where the traveller pulleys represent the wheels of

2736-568: Is thus an equivalence class of directed segments with the same magnitude (e.g., the length of the line segment ( A , B ) ) and same direction (e.g., the direction from A to B ). In physics, Euclidean vectors are used to represent physical quantities that have both magnitude and direction, but are not located at a specific place, in contrast to scalars , which have no direction. For example, velocity , forces and acceleration are represented by vectors. In modern geometry, Euclidean spaces are often defined from linear algebra . More precisely,

2812-409: Is very high, for example typically 30 for a traveller on a steel rope, as visible in aerial photographs, and the other can vary from low, e.g. 1-2 without a centerboard, to 3.5 with one. A diagram is shown which follows the standard force diagram for sailing. It is drawn with a traveller L/D of only about 6 in order to make it clearer. The ferry L/D is drawn at 1.5. The lift L acts at right angles to

2888-617: Is what is needed to "carry" the point A to the point B ; the Latin word vector means "carrier". It was first used by 18th century astronomers investigating planetary revolution around the Sun. The magnitude of the vector is the distance between the two points, and the direction refers to the direction of displacement from A to B . Many algebraic operations on real numbers such as addition , subtraction , multiplication , and negation have close analogues for vectors, operations which obey

2964-413: The n -tuple of its Cartesian coordinates, and every vector to its coordinate vector . Since the physicist's concept of force has a direction and a magnitude, it may be seen as a vector. As an example, consider a rightward force F of 15 newtons . If the positive axis is also directed rightward, then F is represented by the vector 15 N, and if positive points leftward, then the vector for F

3040-487: The coordinates of the endpoint of the vector, with respect to a given Cartesian coordinate system , and are typically called the scalar components (or scalar projections ) of the vector on the axes of the coordinate system. As an example in two dimensions (see figure), the vector from the origin O = (0, 0) to the point A = (2, 3) is simply written as a = ( 2 , 3 ) . {\displaystyle \mathbf {a} =(2,3).} The notion that

3116-417: The dot product . This makes sense, as the addition in such a vector space acts freely and transitively on the vector space itself. That is, R n {\displaystyle \mathbb {R} ^{n}} is a Euclidean space, with itself as an associated vector space, and the dot product as an inner product. The Euclidean space R n {\displaystyle \mathbb {R} ^{n}}

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3192-417: The electric and magnetic field , are represented as a system of vectors at each point of a physical space; that is, a vector field . Examples of quantities that have magnitude and direction, but fail to follow the rules of vector addition, are angular displacement and electric current. Consequently, these are not vectors. In the Cartesian coordinate system , a bound vector can be represented by identifying

3268-396: The hat symbol ^ {\displaystyle \mathbf {\hat {}} } typically denotes unit vectors ). In this case, the scalar and vector components are denoted respectively a x , a y , a z , and a x , a y , a z (note the difference in boldface). Thus, a = a x + a y +

3344-448: The real line , Hamilton considered the vector v to be the imaginary part of a quaternion: The algebraically imaginary part, being geometrically constructed by a straight line, or radius vector, which has, in general, for each determined quaternion, a determined length and determined direction in space, may be called the vector part, or simply the vector of the quaternion. Several other mathematicians developed vector-like systems in

3420-525: The Île Bizard . Reaction ferries cross the rivers Sava and Drava . A number of reaction ferries operate in Germany, particularly across the rivers Elbe and Weser . Between the 17th and 19th centuries, they were quite common on the Rhine . Currently operating ferries include: The Traghetto di Leonardo  [ it ] is a historic reaction ferry across the Adda River at Imbersago . It

3496-491: The 19th century) as equivalence classes under equipollence , of ordered pairs of points; two pairs ( A , B ) and ( C , D ) being equipollent if the points A , B , D , C , in this order, form a parallelogram . Such an equivalence class is called a vector , more precisely, a Euclidean vector. The equivalence class of ( A , B ) is often denoted A B → . {\displaystyle {\overrightarrow {AB}}.} A Euclidean vector

3572-534: The associated vector space (a basis such that the inner product of two basis vectors is 0 if they are different and 1 if they are equal). This defines Cartesian coordinates of any point P of the space, as the coordinates on this basis of the vector O P → . {\displaystyle {\overrightarrow {OP}}.} These choices define an isomorphism of the given Euclidean space onto R n , {\displaystyle \mathbb {R} ^{n},} by mapping any point to

3648-433: The basis has, so the components of the vector must change to compensate. The vector is called covariant or contravariant , depending on how the transformation of the vector's components is related to the transformation of the basis. In general, contravariant vectors are "regular vectors" with units of distance (such as a displacement), or distance times some other unit (such as velocity or acceleration); covariant vectors, on

3724-446: The bridle or multiple tethers in order to steer. The lateral force of the current moves the ferry across the river. A now rare type of reaction ferry uses a submerged cable lying on the bottom across a river or tidal water. This can be a wire rope or a chain and is pulled to the surface by the ferry or its operator. It passes through moveable pulleys or belaying points whose location sets the ferry's angle. In order to set off, manual work

3800-442: The choice of the axes on which the vector is projected. Moreover, the use of Cartesian unit vectors such as x ^ , y ^ , z ^ {\displaystyle \mathbf {\hat {x}} ,\mathbf {\hat {y}} ,\mathbf {\hat {z}} } as a basis in which to represent a vector is not mandated. Vectors can also be expressed in terms of an arbitrary basis, including

3876-441: The complete quaternion product. This approach made vector calculations available to engineers—and others working in three dimensions and skeptical of the fourth. Josiah Willard Gibbs , who was exposed to quaternions through James Clerk Maxwell 's Treatise on Electricity and Magnetism , separated off their vector part for independent treatment. The first half of Gibbs's Elements of Vector Analysis , published in 1881, presents what

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3952-410: The coordinates of its initial and terminal point. For instance, the points A = (1, 0, 0) and B = (0, 1, 0) in space determine the bound vector A B → {\displaystyle {\overrightarrow {AB}}} pointing from the point x = 1 on the x -axis to the point y = 1 on the y -axis. In Cartesian coordinates, a free vector may be thought of in terms of

4028-443: The crossing speed is the same as the speed of the true current. With a centerboard or keel, the hull's L/D could increase several times. This would increase the crossing speed also several times, but according to the drag equation the forces increase with the square of the speed and put a great load particularly on the overhead cable. With the anchored-tether type ferry, such high speeds would be unobtainable because its tether drags in

4104-433: The direction in which the arrow points indicates the vector's direction. On a two-dimensional diagram, a vector perpendicular to the plane of the diagram is sometimes desired. These vectors are commonly shown as small circles. A circle with a dot at its centre (Unicode U+2299 ⊙) indicates a vector pointing out of the front of the diagram, toward the viewer. A circle with a cross inscribed in it (Unicode U+2297 ⊗) indicates

4180-408: The direction of the apparent current, the vector sum of the true current and the current component due to the crossing speed. The drag D acts parallel to the apparent current. The vector sum of L and D is the resultant force R. This force can only exist because the tether exerts an opposed force of the same magnitude (see Newton's laws of motion ), in this simplified two-dimensional projection of what

4256-652: The familiar algebraic laws of commutativity , associativity , and distributivity . These operations and associated laws qualify Euclidean vectors as an example of the more generalized concept of vectors defined simply as elements of a vector space . Vectors play an important role in physics : the velocity and acceleration of a moving object and the forces acting on it can all be described with vectors. Many other physical quantities can be usefully thought of as vectors. Although most of them do not represent distances (except, for example, position or displacement ), their magnitude and direction can still be represented by

4332-401: The ferry to the current. Sometimes two pulleys and tethers are used. Sometimes a single tether is attached to a bar that can be swung from one side of the ferry to the other. This type also uses a rudder in order to set the angle of the ferry to the current flow from zero - it is then stationary - to the best angle for maximal crossing speed. Ferries without a rudder change the relative lengths of

4408-411: The geometrical and physical settings, it is sometimes possible to associate, in a natural way, a length or magnitude and a direction to vectors. In addition, the notion of direction is strictly associated with the notion of an angle between two vectors. If the dot product of two vectors is defined—a scalar-valued product of two vectors—then it is also possible to define a length; the dot product gives

4484-435: The intuitive interpretation as vectors of unit length pointing up the x -, y -, and z -axis of a Cartesian coordinate system , respectively. In terms of these, any vector a in R can be expressed in the form: a = ( a 1 , a 2 , a 3 ) = a 1 ( 1 , 0 , 0 ) + a 2 ( 0 , 1 , 0 ) +

4560-406: The length and direction of an arrow. The mathematical representation of a physical vector depends on the coordinate system used to describe it. Other vector-like objects that describe physical quantities and transform in a similar way under changes of the coordinate system include pseudovectors and tensors . The vector concept, as it is known today, is the result of a gradual development over

4636-520: The middle of the nineteenth century, including Augustin Cauchy , Hermann Grassmann , August Möbius , Comte de Saint-Venant , and Matthew O'Brien . Grassmann's 1840 work Theorie der Ebbe und Flut (Theory of the Ebb and Flow) was the first system of spatial analysis that is similar to today's system, and had ideas corresponding to the cross product, scalar product and vector differentiation. Grassmann's work

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4712-405: The origin as a common base point. A vector a will be written as a = a 1 e 1 + a 2 e 2 + a 3 e 3 . {\displaystyle {\mathbf {a} }=a_{1}{\mathbf {e} }_{1}+a_{2}{\mathbf {e} }_{2}+a_{3}{\mathbf {e} }_{3}.} Two vectors are said to be equal if they have

4788-488: The other hand, have units of one-over-distance such as gradient . If you change units (a special case of a change of basis ) from meters to millimeters, a scale factor of 1/1000, a displacement of 1 m becomes 1000 mm—a contravariant change in numerical value. In contrast, a gradient of 1  K /m becomes 0.001 K/mm—a covariant change in value (for more, see covariance and contravariance of vectors ). Tensors are another type of quantity that behave in this way;

4864-408: The parallelogram). In any dimension (and, in particular, higher dimensions), it is possible to define the exterior product , which (among other things) supplies an algebraic characterization of the area and orientation in space of the n -dimensional parallelotope defined by n vectors. In a pseudo-Euclidean space , a vector's squared length can be positive, negative, or zero. An important example

4940-434: The plane, and thus erected the first space of vectors in the plane. The term vector was introduced by William Rowan Hamilton as part of a quaternion , which is a sum q = s + v of a real number s (also called scalar ) and a 3-dimensional vector . Like Bellavitis, Hamilton viewed vectors as representative of classes of equipollent directed segments. As complex numbers use an imaginary unit to complement

5016-596: The rules of vector addition. An example is velocity , the magnitude of which is speed . For instance, the velocity 5 meters per second upward could be represented by the vector (0, 5) (in 2 dimensions with the positive y -axis as 'up'). Another quantity represented by a vector is force , since it has a magnitude and direction and follows the rules of vector addition. Vectors also describe many other physical quantities, such as linear displacement, displacement , linear acceleration, angular acceleration , linear momentum , and angular momentum . Other physical vectors, such as

5092-563: The same free vector if they have the same magnitude and direction: that is, they are equipollent if the quadrilateral ABB′A′ is a parallelogram . If the Euclidean space is equipped with a choice of origin , then a free vector is equivalent to the bound vector of the same magnitude and direction whose initial point is the origin. The term vector also has generalizations to higher dimensions, and to more formal approaches with much wider applications. In classical Euclidean geometry (i.e., synthetic geometry ), vectors were introduced (during

5168-699: The same magnitude and direction. Equivalently they will be equal if their coordinates are equal. So two vectors a = a 1 e 1 + a 2 e 2 + a 3 e 3 {\displaystyle {\mathbf {a} }=a_{1}{\mathbf {e} }_{1}+a_{2}{\mathbf {e} }_{2}+a_{3}{\mathbf {e} }_{3}} and b = b 1 e 1 + b 2 e 2 + b 3 e 3 {\displaystyle {\mathbf {b} }=b_{1}{\mathbf {e} }_{1}+b_{2}{\mathbf {e} }_{2}+b_{3}{\mathbf {e} }_{3}} are equal if

5244-412: The tail of the vector coincides with the origin is implicit and easily understood. Thus, the more explicit notation O A → {\displaystyle {\overrightarrow {OA}}} is usually deemed not necessary (and is indeed rarely used). In three dimensional Euclidean space (or R ), vectors are identified with triples of scalar components: a = (

5320-487: The two definitions of Euclidean spaces are equivalent, and that the equivalence classes under equipollence may be identified with translations. Sometimes, Euclidean vectors are considered without reference to a Euclidean space. In this case, a Euclidean vector is an element of a normed vector space of finite dimension over the reals, or, typically, an element of the real coordinate space R n {\displaystyle \mathbb {R} ^{n}} equipped with

5396-702: The unit vectors of a cylindrical coordinate system ( ρ ^ , ϕ ^ , z ^ {\displaystyle {\boldsymbol {\hat {\rho }}},{\boldsymbol {\hat {\phi }}},\mathbf {\hat {z}} } ) or spherical coordinate system ( r ^ , θ ^ , ϕ ^ {\displaystyle \mathbf {\hat {r}} ,{\boldsymbol {\hat {\theta }}},{\boldsymbol {\hat {\phi }}}} ). The latter two choices are more convenient for solving problems which possess cylindrical or spherical symmetry, respectively. The choice of

5472-592: The water or is supported by buoys that do and this drag would also increase with the square of the speed. At one time over 30 reaction ferries crossed the rivers of British Columbia , primarily the Fraser River and the Thompson River . Those still operating include: In Quebec , the small Laval-sur-le-Lac–Île-Bizard Ferry operates seasonally across the Rivière des Prairies from Laval-sur-le-Lac to

5548-409: The water. Such ferries operate faster and more effectively in rivers with strong currents. Some reaction ferries operate using an overhead cable suspended from towers anchored on either bank of the river at right angles to the current flow. A "traveller" with pulleys runs along this cable and is attached to the ferry with a tether rope. This can divide into a two-part bridle which defines the angle of

5624-496: Was constructed by the National Park Service in 2009. Euclidean vector#Addition and subtraction In mathematics , physics , and engineering , a Euclidean vector or simply a vector (sometimes called a geometric vector or spatial vector ) is a geometric object that has magnitude (or length ) and direction . Euclidean vectors can be added and scaled to form a vector space . A vector quantity

5700-407: Was largely neglected until the 1870s. Peter Guthrie Tait carried the quaternion standard after Hamilton. His 1867 Elementary Treatise of Quaternions included extensive treatment of the nabla or del operator ∇. In 1878, Elements of Dynamic was published by William Kingdon Clifford . Clifford simplified the quaternion study by isolating the dot product and cross product of two vectors from

5776-664: Was operated partly by the current and partly by punting . Several reaction ferries crossed rivers in the Ozark Mountains of the central United States during the first half of the 20th century. The Akers Ferry across the Current River near Salem in Missouri remains in operation. Menor's Ferry in Jackson Hole, Wyoming, was a dual-pontoon reaction ferry built in the 1890s and operated until 1927. A replica

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